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A Blackjack Question?

5 August 2012

By Donald Catlin

I recently received an email from one of my readers, Amy, posing a question about Blackjack. Here is Amy's email:

Dear Sir,

My name is Amy, and I only ever play blackjack, basic strategy, but not card counting. My question is, regardless how long I have to play … minutes or hours … what are my chances of winning just two hands … not necessarily consecutively … meaning I bet say $50 and win $100 … with an unlimited bankroll … then walk away … do I have more chance of walking away a winner every time.

Thanks,
Amy

First of all Amy, thank you for writing to me. Your email brings up some interesting points. To illustrate them I would like to critique your email and then pose the question that I think you should be asking.

In a one-dimensional random walk (which is what a gambler's stake is), the probability of success is the probability of ruin subtracted from 1. Since you postulate an unlimited bankroll the probability of ruin is zero hence one would conclude that success is certain. However, since you are playing a negative game it is certainly possible that one could lose more and more as play continues and never reach the desired goal. In other words, postulating an unlimited bankroll can lead to an erroneous conclusion.

My second criticism is in the sentence fragment "do I have more chance of walking away a winner every time." More chance than what? This is too vague and I don't know how to answer this.

Amy I think you want to know what your chances of success are when starting with a large stake and playing for modest gains. For example, let's say you start with $1000 and, making $50 bets, play to win $100. If you go back to my October 2011 article I derive a formula for approximating Blackjack with a one-dimensional random walk with a fixed step size (a fixed wager). Here are the numbers you need.

We might as well start with 20 units and play for 2, it's the same situation. So using the notation in my article:

S= 1.1225, M = -0.0058, B = 20, G = 2, and w = 1

Using these numbers the ruin probability turned out to be 0.144778471 so the probability of success is 0.855221528. You can adjust B to be a larger number but make sure that your calculator has accuracy to many decimal places. Best wishes Amy and good luck playing Blackjack. See you all next month.


Don Catlin can be reached at 711cat@comcast.net

Donald Catlin
Don Catlin is a retired professor of mathematics and statistics from the University of Massachusetts. His original research area was in Stochastic Estimation applied to submarine navigation problems but has spent the last several years doing gaming analysis for gaming developers and writing about gaming. He is the author of The Lottery Book, The Truth Behind the Numbers published by Bonus books.

Books by Donald Catlin:

Lottery Book: The Truth Behind the Numbers